#### Transcript Class06

```How To Say What You Want
Describing Signals
Review
• Light is trapped in an optical fiber if it strikes the
sides of the fiber at angles greater than the critical
• The core must have a higher index of refraction than
the cladding for total internal reflection to occur.
• The numerical aperture (NA) of a fiber relates the
maximum angle of incidence on the front of the
fiber to the indices of refraction of the fiber:
NA = n0 sin qm = (n12 - n22)1/2.
Review (cont.)
• Any periodic function of frequency f0 can be expressed as a
sum over frequency of sinusoidal waves having frequencies
equal to nf0, where n is an integer. The sum is called the
Fourier series of the function, and a plot of amplitude
(coefficient of each sin/cos term) vs. frequency is called the
Fourier spectrum of the function.
• Any non-periodic function (so frequency f0 0) can be
expressed as an integral over frequency of sinusoidal waves
having frequencies. The integral is called the Fourier
transform of the function, and a plot of amplitude vs.
frequency is called the Fourier spectrum of the function.
• The Fourier spectrum of a wider pulse will be narrower than
that of a narrow pulse, so it has a smaller bandwidth.
How do we send signals?
• Radio antenna (AM frequencies around 1000 kHz,
FM frequencies around 100 MHz)
• TV antenna (VHF frequencies are around 100 MHz,
on either side of FM frequencies, UHF frequencies
around 500 MHz)
These are public transmissions, and so the carrier
frequencies are set and regulated
• Coaxial cable
These are private
• Optical waveguides
transmissions, and sent
over range of frequencies
• ISDN
Optical waveguides pros and cons
• Message remains private
• Flexibility
• Low Loss
• Insensitive to EM interference
• Very high bandwidth
BUT
• Expensive to connect to every house
• Require electricity-to-light converters
• Either multi-modal, or less efficient (better coupling makes
this less relevant)
Chromatic Dispersion
• Index of refraction is dependent on wavelength.
• Typical materials exhibit higher indices of
refraction for lower wavelengths (higher energies)
• Thus violet light bends the most through a prism
or water and appears on the outside of a rainbow.
Optical Fiber
Dispersion/Attenuation
• Signals in a fiber will have several sources of dispersion:
– Chromatic:
• Material: index of refraction depends on wavelength (prism)
• Waveguide: some of wave travels through cladding with
different index of refraction (primarily single-mode) – leads to
wavelength-dependent effects
– Modal: different modes travel different paths and so
require different amounts of time to travel down fiber
(CUPS)
• Also have attenuation/loss due to scattering/absorption by
fiber material, which depends on wavelength/frequency
Modes in Optical Fibers
• Modes in a fiber are specific field
distributions that are independent of the
length traveled down the fiber. However,
these modes travel at slightly different
speeds which matters more with longer
fibers.
• Fields of modes look like harmonics of
standing waves
Modes Combine to Give Path of Light
• BUT, modes travel at different speeds, so sum of fields changes as
go down the fiber
• Result is one of the paths light will take
Mode 1
Mode 2
2
Intensity Pattern of Sum
• (Figures adapted from Photonics – not to scale)
Modes in Optical Fibers (cont.)
• Modes are like sines and cosines in Fourier
series: each path that light takes down a
fiber can be expressed as a sum of modes.
• Can make a single-mode fiber by:
– reducing diameter of fiber so smaller cone of
light enters
– reducing NA of fiber so smaller cone of light is
trapped
What Exactly Is Bandwidth, and
Why Do We Care?
• A range of frequencies
• Generally found by taking the frequencies with amplitudes more
than half the maximum amplitude (e.g., on a Fourier spectrum)
• Bandwidth for a medium is the range of frequencies which can
pass through that medium with a minimum of separation
• Sampling theory says that a signal transmitting N different
amplitudes per second requires a bandwidth of at least N/2:
B>N/2
• Usually this ideal is not achieved, and the required bandwidth is
larger
– Grant says B approx N
Some ways to modulate signals
• Amplitude modulation:
– A signal with a constant carrier frequency is sent
– The original signal becomes the amplitude of the transmitted signal
– Since the transmitted signal is not a simple sine wave, it has a
bandwidth of Fourier components
• Frequency modulation:
– A signal with a constant carrier frequency is sent
– The original signal becomes the change in frequency of the
transmitted signal
– Since the transmitted signal is not a simple sine wave, it has a
bandwidth of Fourier components
– FM is easier to amplify, since only the frequency determines the
signal.
Schemes for Encoding
• ASCII - The American Standard Code for Information
Interchange is a standard seven-bit code that was
proposed in 1963, and finalized in 1968. ASCII was
established to achieve compatibility between various
types of data processing equipment.
• ASCII is the common code for microcomputer
equipment. The standard ASCII character set consists of
128 decimal numbers ranging from zero through 127
assigned to letters, numbers, punctuation marks, and the
most common special characters. The Extended ASCII
Character Set also consists of 128 decimal numbers and
ranges from 128 through 255.
Pulses and Data
• Can represent binary data with pulses in a variety of ways
• 10110 could look like . . .
Notice that the NRZ
takes half the time of
the others for the
same pulse widths
Non-return-to-zero
(NRZ)
Manchester
Coding
Return-to-zero
(RZ)
Bipolar Coding
Distortion
• No physical change is instantaneous
• If change is too slow, won’t have time to rise before needs to fall
• Results in data loss
Sharp edges
Sizeable rise time
Really Distorted
• Since rise is generally exponential, we define “rise time” to
be time from 10% of max value to 90% of max; “fall time”
is time from 90% to 10%
• To be able to resolve data, the rise time and fall time must
be less than 70% of the bit width
What exactly is a decibel?
• A ratio, often of power
• BUT, in logarithmic form:
• dB = 10 log (P2/P1)
• e.g., if my received signal is 1/10 as big as my transmitted
signal, my “gain” would be
gain dB = 10 log (1/10) = -10
• The minus sign denotes loss, or a second power less than the
initial power
• Signal-to-noise ratios are often given in decibels
• You want the signal to be larger than the noise, so the ratio
(in dB) should be positive
• For digital data, we use bit error rate, not signal-to-noise
• Bit error rate is ratio of wrong bits to total bits - it should be
small, whereas SNR should be large
• Bit error rate can be expressed as a plain number, or in
decibels
Before the next class, . . .